How do I set up a control bone for a mechanism that follows an arc?
You could achieve that using an action constraint on the arc bone.

First create an animation of the lift going from the lowest position to the highest position (or vice versa) and then set the z axis location of the control bone to be what drives the action. you'll need to adjust the settings of the action constraint to fit the length of your animation and define the range as well.

If your goal is to have the platform move along the z-axis at a constant rate (which is what you would need for it to always be at the level of the control) you could use a circular interpolation for your animation so the change in height would be linear.

However, doing this would result in rapid rotation of the arc bone at the top and bottom of the lift, which may or may not be a desired result.